1 Introduction

      I assert . . . that in any special doctrine of nature there can be only as much proper science as there is mathematics therein. For . . . proper science, and above all proper natural science, requires a pure part lying at the basis of the empirical part.

      —Immanuel Kant

      In the twentieth and twenty-first centuries, there has been a substantial expansion in the number of fields applying mathematics to their investigations of natural and social phenomena. Areas of social research such as psychology and sociology, which had traditionally been qualitative, have developed robust quantitative components, such as standardized intelligence tests and statistical surveys to assess the habits, beliefs, and practices of populations. In addition, fields of natural investigation. particularly genetics and evolutionary biology, have expanded their methods as described in the Forward from observation and experiment to include mathematical descriptions of the genome and the change and distribution of variation within organic populations. As a result of this expansion, mathematics has become a ubiquitous aspect of what makes a discipline “scientific,” making Kant’s invocation that “there can only be as much proper science as there is mathematics therein” seem even more relevant today as it was when he wrote it in 1786.1

      Although this mathematical expansion has helped us better understand social, psychological, and biological phenomena, the path to developing and adopting mathematics in science has not always straight or easy. One widely-cited example of a mathematical science with a turbulent beginning is population genetics. In the twenty- first century, this vital field of genetics research has its own textbooks, specialists, and places in the academy. Its recent success, however, obscures a less flourishing past. Although the basic scientific and mathematical foundations for population genetics were largely in place by the first decade of the twentieth century, almost thirty years elapsed before it garnered sufficient attention and support to establish itself as a field of research worthy of an individuated identity.

      Historians interested in the intellectual foundations of population genetics and its transition into an important field of inquiry have reached back to Darwin and traced its development forward, hoping to understand the reasons why such a productive, modern field of study had such a difficult maturation. Perhaps the most well-known investigation into this mystery was undertaken by historian William Provine, whose groundbreaking book, The Origins of Theoretical Population Genetics, suggests that the turmoil associated with the rise of the field was largely the consequence of an ideological conflict between Darwinians and Mendelians (ix-x).

      According to Provine, Darwin and his later followers, the biometricians, believed in continuous variation in which differences between members of a species arose by the slow accretion of small variations over long periods of time. Mendelians and the supporters of mutation theory, on the other hand, believed that variation was discontinuous: varieties appeared suddenly and could introduce dramatic changes into individuals and populations of organisms. By tracing these notions about variation from Darwin through the debates between the biometricians and the Mendelians, Provine concludes that it was only when the differences between these ideological positions were resolved in the work of R.A. Fisher, J.B.S. Haldane, and Sewall Wright, that a research field of population genetics emerged (Provine 131).

      Although Provine and other historians rightfully devote attention to how ideological conflicts over variation complicated the development of population genetics, their focus excludes other elements that could have contributed significantly to population genetics’ “torturous” development (Provine ix). One important element that has not been considered is whether and to what degree the beliefs about the acceptability of using mathematics to make arguments about biological phenomena might have contributed to the difficulties in establishing the field.

      Although it is difficult from a twenty-first century perspective to image mathematics not being a legitimate means of researching variation, evolution, and heredity, scrutiny of the work of early researchers such Charles Darwin, Gregor Mendel, Francis Galton, Karl Pearson, and R.A Fisher suggests that this has not always been the case. Attention to their work and its reception reveals that mathematical approaches to these phenomena were caught up in a cycle of development, conflict, and persuasion that lasted almost one hundred years, a cycle that has all but been forgotten as science looks to the future and eviscerates from memory the useless, blind allies and conflicts that led to its current position. In hoping to understand the development of scientific knowledge, however, we need to look at the process of making knowledge, not just the results. Investigating these long-forgotten conflicts can help us understand not only what people believed, but also how they were moved to change their beliefs, what they perceived were good reasons for accepting or rejecting a particular position, and what lines of argument dominated the scientific landscape.

      Despite the importance of mathematics to scientific argument and epistemology, there have been few historical-philosophical, sociological, or rhetorical investigations of how scientists argue with or about the use of mathematics.2 In the history of science, the intersection between argument, science, and mathematics has been investigated by historian Peter Dear, whose book, Discipline and Experience: The Mathematical Way in the Scientific Revolution, examines sixteenth, seventeenth, and eighteenth century disputes among natural philosophers about whether mathematics could serve legitimately and authoritatively as a source for arguments about nature. In the book, Dear attempts to make sense—in the context of eighteenth century natural philosophy—of both the novelty of Newton’s physico-mathematical argument strategy and its importance to the development of a new paradigm for scientific research (248).

      Although very few historians have examined the relationships between mathematics, science, and argument, there is evidence of a trend towards increasing attention to the subject. In a recent discussion, for example, in Isis—a top journal in the study of history and philosophy of science—titled, “Ten Problems in the History and Philosophy of Science,” historian and philosopher Peter Galison lists a lack of understanding of the “Technologies of Argumentation” as problem number three for philosophers and historians of science. He raises the following questions for them to pursue:

      When the focus is on scientific practices (rather than discipline-specific scientific results per se), what are the concepts, tools, and procedures needed at a given time to construct an acceptable scientific argument? . . . Cutting across subdisciplines and even disciplines, what is the toolkit of argumentation and demonstration—and what is its historical trajectory? (116)

      For rhetoricians of science, whose interests lay predominantly in the study of scientific argument and communicative practices, answering these sorts of questions about the relationship between mathematics, science, and argument would seem to be an important and fruitful undertaking. Despite the natural fit between scholarly interest and subject matter, however, very few rhetoricians have made efforts to examine the intersection between these three subjects. One notable exception is the work of Alan Gross, Joseph Harmon, and Michael Reidy in, Communicating Science: The Scientific Article from the 17th Century to the Present. In this book, the authors examine the developing conventions of argument and style in the scientific article, including brief descriptions of the use of mathematics.

      Though the works of Dear, Gross, Harmon and Reidy begin a conversation about the role of mathematics in scientific argument, there are many important avenues currently unexplored. Questions—such as, “Do new mathematical methods have a different status of reliability as a source for arguments in science than existing ones?”; “If mathematical methods are not assumed a priori to be reliable, how do scientists